# Does every random variable(continous) has a probability density function?

what is the criterion for a random variable(continous) for existence of probability density function for it? Could you provide some cases of random variable(continous) where pdf ceases to exist.

• How are you defining a continuous random variable? – M. Vinay Jun 19 '14 at 5:17
• Sometimes "continuous random variable" means "random variable with a density function." – Qiaochu Yuan Jun 19 '14 at 5:18
• a continous random variable has continous CDF function. – VKV Jun 19 '14 at 5:18
• It happens if the cdf is absolutely continuous. – André Nicolas Jun 19 '14 at 5:20
• An example in which it is not absolutely continuous is en.wikipedia.org/wiki/Cantor_distribution – M. Vinay Jun 19 '14 at 5:22

A (real-valued) random variable $X$ has density $f$ if you can write $$P(X \leq x) = \int_{-\infty}^x f(y) dy$$.